The weight spectrum of two families of Reed-Muller codes

Abstract

We determine the weight spectra of the Reed-Muller codes RM (m− 3, m) for m ≥ 6 and RM (m − 4, m) for m ≥ 8. The technique used is induction on m, using that the sum of two weights in RM (r −1, m−1) is a weight in RM (r, m), and using the characterization by Kasami and Tokura of the weights in RM (r, m) that lie between its minimum distance 2m−r and the double of this minimum distance. We also de- rive the weights of RM (3, 8), RM (4, 9), by the same technique. We conclude with a conjecture on the weights of RM (m − c, m), where c is fixed and m is large enough.